evaluate lim x tends a f(x) where f(x) = (ax- xa)/( ax - xx)

Given  y = (ax - xa) / (ax - xx )

As limit a → 0 , y = 0/0 which is indeterminate

Now differentiating numerator and denominator separately we get

d(xa)/dx = axa-1

for xx

Let k = xx

ln k = x ln x

Differentiating both sides we get

(1/k)dk/dx = (ln x )(dx/dx) + (d (lnx)/dx) (x)  = ln x + (1/x)x = 1+ ln x

d(ax)/dx = ax(ln a)

y = lim ((ln a)ax - axa-1) / ((ln a)ax - (1+ln x )xx)

⇒ y =  ((ln a)aa - aaa-1) / ((ln a)aa - (1+ln a )aa)

⇒ y =  ((ln a)aa - aa) / ((ln a)aa - (1+ln a )aa)  

= aa(ln a -1)/(aaln a -aa -aa lna) 

= aa(ln a -1)/(-aa

= 1- ln a

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